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Keiji Okano

Publications and source records attributed to Keiji Okano.

5 recordsLinked to original sources
arXiv2026

On the triviality of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions

For a number field $k$ and an odd prime number $p$, we consider the maximal multiple $\mathbb{Z}_p$-extension $\tilde{k}$ of $k$ and the unramified Iwasawa module $X(\tilde{k})$, which is the Galois group of the maximal unramified abelian $p$-extension of $\tilde{k}$. In this article, we classify the CM-fields $k$ in which $p$ splits completely and for which $X(\tilde{k}) = 0$. To this end, we provide a relation between the trivialities of $X(\tilde{k})$ and that of the $p$-split Iwasawa modules associated with certain anticyclotomic $\mathbb{Z}_p$-extensions. In addition, we provide an alternative proof of the sufficient condition for $X(\tilde{k})=0$, based on the study of the generalized Greenberg conjecture.

math.NT↗
arXiv2021

On the Cyclicity of the Unramified Iwasawa Modules of the Maximal Multiple $\mathbb{Z}_p$-Extensions Over Imaginary Quadratic Fields

For an odd prime number $p$, we study the number of generators of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg's generalized conjecture holds.

math.NT↗
arXiv2019

Galois coinvariants of the unramified Iwasawa modules of multiple $\mathbb{Z}_p$-extensions

For a CM-field $K$ and an odd prime number $p$, let $\widetilde K'$ be a certain multiple $\mathbb{Z}_p$-extension of $K$. In this paper, we study several basic properties of the unramified Iwasawa module $X_{\widetilde K'}$ of $\widetilde K'$ as a $\mathbb{Z}_p[[{\rm Gal}(\widetilde K'/K)]]$-module. Our first main result is a description of the order of a Galois coinvariant of $X_{\widetilde K'}$ in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic $\mathbb{Z}_p$-extension of $K$ under a certain assumption on the splitting of primes above $p$. Second one is that if $K$ is an imaginary quadratic field and $p$ does not split in $K$, we give a necessary and sufficient condition for which $X_{\widetilde K}$ is $\mathbb{Z}_p[[{\rm Gal}(\widetilde K/K)]]$-cyclic under several assumptions on the Iwasawa $λ$-invariant and the ideal class group of $K$, where $\widetilde K$ is the $\mathbb{Z}_p^2$-extension of $K$.

math.NT↗
arXiv2016

Note on families of pairing-friendly elliptic curves with small embedding degree

Pairing-based cryptographic schemes require so-called pairing-friendly elliptic curves, which have special properties. The set of pairing-friendly elliptic curves that are generated by given polynomials form a complete family. Although a complete family with a $ρ$-value of 1 is the ideal case, there is only one such example that is known, this was given by Barreto and Naehrig. We prove that there are no ideal families with embedding degree 3, 4, or 6 and that many complete families with embedding degree 8 or 12 are nonideal, even if we chose noncyclotomic families.

cs.CR↗